Preview

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Chapter 7 Wavelets and Multiresolution Processing

• Preview
• What is multi-resolution?
• The difference between Fourier transform and
  Wavelet transform
• 7.1 Background
• Both small and large objects, or low and high
  contrast objects are present
• Examine an object --Depending on the size or
  contrast of the object
• Local histogram variations (Fig. 7.1)
• 7.1.1 Image Pyramids
• What is an image pyramid?

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• Block diagram for image pyramid
• Approximation pyramid
• Prediction pyramid
• Matrix pyramids
• a sequence of images are used when it is
  necessary to work with image at different
  resolutions simultaneously
• Tree pyramids
• use several resolutions simultaneously

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Chapter 7 Wavelets and Multiresolution Processing
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• Quad-trees
• Modifications of T-pyramids
• Every node of the tree except the leaves has four
  children
• The image is divided into four quadrants at each
  hierarchical level
• If a parent node has four children if the same
  value, it is not necessary to record them
• Matrix pyramids
• The total number of elements in a P1 level
  pyramid
• Approximation pyramid
• Prediction residual pyramid
• An image with level J and its P reduced
  resolution
• Contains a low-resolution approximation of the
  original at level J-P and information for the
  construction of P higher-resolution approximation
  at the other level

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• A PI level pyramid is built by executing the
  operations in the block diagram P times
• first iteration produces the level J-1
  approximation and level J residual results
• each pass is composed of three steps (Fig.
  7.2(b))
• Step 1 compute a reduced-resolution
  approximation of the input imagefiltering and
  down-sampling
• Mean pyramid, low-pass Gaussian filter based on
  Gaussian pyramid, no filtering (i.e.sub-sampling
  pyramid)
• If we compute without filtering, alias can become
  pronounced
• Step 2
• 1. up-sample the o/p of the step (a)-again
  by a factor of 2. filter--interpolate intensities
  between the pixels of the step 1
• Create a prediction image
• Determines how accurately approximate the input
  by using interpolation
• If we delete interpolation filter, blocky effect
  is inevitable

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• Step3 compute the difference between the
  prediction of step2 and the input to step 1
  (prediction residual)
• Predict residual of level J
• Can be used to reconstruct the original image
• Can be used to generate the corresponding
  approximation pyramid including the original
  image without quantization error
• level j-1 approximation can be used to populate
  the approximation pyramid
• coarse to fine strategy
• High resolution pyramidused for analysis of
  large structure or overall image context
• Low resolution pyramid analyzing individual
  object characteristics

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• the level j prediction residual outputs are
  placed in the prediction residual pyramid
• Ex. Fig. 7.3 (P3)
• Approximation pyramid--Gaussian pyramid (5x5
  low-pass Gaussian kernel)
• Prediction residual--Laplacian pyramid
• 64x64 Laplacian pyramid predict the Gaussian
  pyramids level 7 prediction residual
• First order statistics of the pyramid are highly
  peaked around zero

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Chapter 7 Wavelets and Multiresolution Processing
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Chapter 7 Wavelets and Multiresolution Processing
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7.1.2 Sub-band coding

• An image is decomposed into a set of band-limited
  component sub-bands, which can be reassemble to
  reconstruct the original image
• Each sub-band is generated by band-pass filtering
  its I/p
• the sub-band can be down sampled without loss of
  information
• Reconstruction of the original image is
  accomplished by sampling, filtering, and summing
  the individual sub-band
• The principal components of a two-band sub-band
  coding and decoding system (Fig. 7.4)
• The output sequence is formed through the
  decomposition of x(n) into y0(n) and y1(n) via
  analysis filter h0(n) and h1(n),and subsequent
  recombination via synthesis filters g0(n) and
  g1(n)

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Chapter 7 Wavelets and Multiresolution Processing
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• Bio-orthogonal- filter bank satisfying the
  conditions in (Eq.7.1-21)
• Filter response of two-band, real coefficient,
  perfect reconstruction filter bank are subject to
  bio-orthogonality constraints
• Orthonormal
• Two-dimensional four-band filter bank for subband
  image coding (with one-dimensional filter in
  Table1)
• A four-band split of the 512x512 image , based on
  the filters in Fig. 7.6
• 7.1.3 The Harr transform
• Basic functions are the oldest and simplest known
  orthnormal wavelet
• Separable and symmetric and can be expressed in
  matrix form THFH

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Chapter 7 Wavelets and Multiresolution Processing
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Chapter 7 Wavelets and Multiresolution Processing
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• The Harr basic functions are
• Discrete wavelet transform using Harr basic
  functions
• 7.2 Multiresolution Expansions
• 7.2.1 Series expansions
• A signal f(x) can be analyzed as a linear
  combination of expansion function
• Closed span of the expansion set, denoted
• Three cases using vectors in two-dimensional
  Euclidean space
• 7.2.2 Scaling functions the shape of ?(x)
  changes with j
• Expansion functions composes of integer
  translation. And binary scaling of the real,
  square-integrable function ?(x)

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• The norm of f(x) f(x), is denoted as the
  square root theinner product of f(x) with itself
• L2 denotes the set of measurable,
  square-integrable one-dimensional functions( R
  the set of real numbersZ the set of integers)

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Chapter 7 Wavelets and Multiresolution Processing
Fig. 7.24 (Cont)
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Chapter 7 Wavelets and Multiresolution Processing